On Multiple Completeness of the Root Functions of the Nonregular Pencils of Differential Operators with Constant Coefficients and Splitting boundary Conditions

In the space of square summable functions on the main segment $[0,1]$, the class of polynomial pencils of ordinary differential operators of the $n$-th order is considered. The coefficients of the differential expression are assumed to be constants. The boundary conditions are assumed to be splitting and two-point at the ends 0 and 1 ($l$ of boundary conditions are taken only at the point 0, and the remaining $n−l$ at point 1). The differential expression and the boundary forms are assumed to be homogeneous, that is, they contain only main parts. It is supposed that roots of the characteristic equation of the pencils of this class are simple, non-zero and lie on two rays emanating from the origin in quantities $k$ and $n − k$. Sufficient conditions for $m$-fold completeness (with a possible finite defect) of the system of root functions of the pencils of this class in the space of square integrable functions on the main segment are formulated and proved. In the case of $l \leq min \left \{ k, n-k \right \} $ sufficient conditions of $2l$-fold completeness are proved, and in a case $l \geq max\left \{ k, n-k \right \}$ sufficient conditions of $2(n − l)$-fold completeness are proved. These sufficient conditions consist in difference from zero some quite concrete determinants, constructed on coefficients of boundary conditions and the roots of the characteristic polynomial. Upper bounds are given for possible finite defects. The proof is carried out according to a somewhat modernized “classical” scheme of the proof of completeness (going back to the works of M. B. Keldysh, A. P. Khromov, A. A. Shkalikov and others). In the remainingcase,when $min \left \{ k, n-k \right \} < l < max \left \{ k, n-k \right \}$ ,the $(n−k)$-fold completenessof the root function system is established. In this case, the “method of generating functions” (proposedearlier by the author) is used. This method consists in use instead of “classical” generating functions for the system of root functions introduced by the author of the new “generalized generating functions” depending on arbitrary vector of parameters, and in selection of these vectors of parameters so that classic proof scheme still works.

Keywords: pencil of ordinary differential operators, polynomial pencil of differential operators, homogeneous differential expression, homogeneous boundary forms, multiple completeness, root functions, eigen- and associated functions, derived chains, splitting boundary conditions.